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Proportional approval voting

From Academic Kids

Proportional approval voting (PAV) is a theoretical voting system for multiple-winner elections, in which each voter can vote for as many or as few candidates as the voter chooses. It was developed by Forest Simmons in 2001.

PAV works by looking at how "satisfied" each voter is with each potential result or outcome of the of the election.
The satisfaction for individual voters of a potential result is calculated based on how many of the successful candidates they voted for. In this particular system, if an individual voted for n successful candidates (and an irrelevant number of unsuccessful ones) then their satisfaction is taken to be

<math>1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}<math>

Adding up the satisfaction of all the voters with the potential result gives the total satisfaction with that result. The potential result with the highest total satisfaction is chosen as the actual result.

Without the weighting of satisfaction, i.e. if the numbers of votes for each candidate are simply added up and those with the highest numbers elected, equivalent to satisfaction being n, then this would amount to block approval voting which could have a similar chance of landslide results as block voting.

Proportional approval voting is a computationally complex method of vote counting. If there were c candidates and w winners, then there would be

<math>\frac{c!}{w! (c-w)!}<math>

potential results to compare with each vote. If there were 20 candidates for 5 seats then there would be more than 15,000 potential results. Such elections could only reasonably be counted by computer.

A somewhat simpler counting method is sequential proportional approval voting where candidates are elected one-by-one to the winners' circle by approval voting, but in each round the value of the votes of each voter who already has m candidates in the winners' circle is reduced to

<math>\frac{1}{m+1}<math>

This was developed by the Danish polymath Thorvald N. Thiele, and used (with adaptations) in Sweden for a short period after 1909.